Fig. 1.

These plots constitute the main results of the paper and demonstrate the existence of a potential-gradient-induced MBL phase. (A) The $r$-index as a function of disorder and field strength as calculated for the Hamiltonian in Eq. 5 with $L=16$ and $J_0=J_z=1$ (averaged over 125 realizations). Evidently, a phase boundary exists between a region with $r=0.53$ (Wigner–Dyson) for small values of $W$ and $F$ (the ergodic dome) to a region with $r=0.386$ (Poisson). (B) The averaged participation ratio ($PR=1/IPR$) as a function of disorder and field strength for the same system as in A. Consistent with the level statistics, inside the ergodic dome the $PR$ is proportional to the Hilbert space dimension ($\mathcal{D}$), while outside the dome it becomes small and independent of $\mathcal{D}$. Notice that in B the line $W=0$ is included in the data. In both cases the red line serves only as a guide to eye and is a contour or $r\approx 0.46$.